History

The compass in this 13th-century manuscript is a symbol of God's act of Creation.

Notice also the circular shape of the halo

The word "circle" derives from the Greek κίρκος (kirkos), itself a metathesis of

the Homeric Greek κρίκος (krikos), meaning "hoop" or "ring". The origins of the words

"circus" and "circuit" are closely related.

Circular piece of silk with Mongol images

Circles in an old Arabic astronomical drawing.

The circle has been known since before the beginning of recorded history. Natural circles

would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind

on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with

related inventions such as gears, makes much of modern machinery possible. In mathematics, the

study of the circle has helped inspire the development of geometry, astronomy, and calculus.

DEFINITION OF CIRCLES

A circle is a simple shape of Euclidean geometry that is the set of all points in a plane

that are at a given distance from a given point, the centre. The distance between any of the points

and the centre is called the radius. It can also be defined as the locus of a point equidistant from a

fixed point.

A circle is a simple closed curve which divides the plane into two regions: an interior and

an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the

boundary of the figure, or to the whole figure including its interior; in strict technical usage, the

circle is the former and the latter is called a disk.

A circle can be defined as the curve traced out by a point that moves so that its distance

from a given point is constant.

A circle may also be defined as a special ellipse in which the two foci are coincident and

the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter,

using calculus of variations

Properties

The circle is the shape with the largest area for a given length of perimeter. (Isoperimetric

inequality.)

The circle is a highly symmetric shape: every line through the centre forms a line of

reflection symmetry and it has rotational symmetry around the centre for every angle. Its

symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle

group T.

All circles are similar.

o A circle's circumference and radius are proportional.

o The area enclosed and the square of its radius are proportional.

o The constants of proportionality are 2π and π, respectively.

The circle which is centred at the origin with radius 1 is called the unit circle.

o Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.

Through any three points, not all on the same line, there lies a unique circle. In Cartesian

coordinates, it is possible to give explicit formulae for the coordinates of the centre of the

circle and the radius in terms of the coordinates of the three given points.

Terminology

Arc: any connected part of the circle.

Centre: the point equidistant from the points on the circle.

Chord: a line segment whose endpoints lie on the circle.

Circular sector: a region bounded by two radii and an arc lying between the radii.

Circular segment: a region, not containing the centre, bounded by a chord and an arc

lying between the chord's endpoints.

Circumference: the length of one circuit along the circle.

Diameter: a line segment whose endpoints lie on the circle and which passes through the

centre; or the length of such a line segment, which is the largest distance between any

two points on the circle. It is a special case of a chord, namely the longest chord, and it is

twice the radius.

Passant: a coplanar straight line that does not touch the circle.

Radius: a line segment joining the centre of the circle to any point on the circle itself; or

the length of such a segment, which is half a diameter.

Secant: an extended chord, a coplanar straight line cutting the circle at two points.

Semicircle: a region bounded by a diameter and an arc lying between the diameter's

endpoints. It is a special case of a circular segment, namely the largest one.

Tangent: a coplanar straight line that touches the circle at a single point.

Mathematical Constant π

Chord, secant, tangent, radius, and diameterArc, sector, and segment

CIRCUMFERENCE OF A CIRCLE

Circumference (from Latin circumferentia, meaning "to carry around") is the linear

distance around the edge of a closed curve or circular object.[1] The circumference of a circle is of

special importance in geometry and trigonometry. However "circumference" may also refer to

the edge of elliptical closed curve. Circumference is a special case of perimeter in that the

perimeter is typically around a polygon while circumference is around a closed curve.

Circle illustration with circumference (C) in black, diameter (D) in cyan, radius (R) in red, and centre or

origin (O) in magenta. Circumference = π × diameter = 2 × π × radius.

The circumference of a circle is the distance around it. The term is used when measuring

physical objects, as well as when considering abstract geometric forms.

When a circle's radius is 1, its circumference is 2π.

When a circle's diameter is 1, its circumference is π.

Relationship with Pi

The circumference of a circle relates to one of the most important mathematical constants

in all of mathematics. This constant, pi, is represented by the Greek letter π. The numerical value

of π is 3.14159 26535 89793 ... , and is defined by two proportionality constants. The first

constant is the ratio of a circle's circumference to its diameter and equals π. While the second

constant is the ratio of the diameter and two times the radius and is used as to convert the

diameter to radius in the same ratio as the first, π. Both proportionality constants combine in

respect with circumference c, diameter d, and radius r to become:

The use of the mathematical constant π is ubiquitous in mathematics, engineering, and science.

While the constant ratio of circumference to radius also has many uses in

mathematics, engineering, and science, it is not formally named. These uses include but are not

limited to radians, computer programming, and physical constants.

Example-1

What is the circumference of the circle?

Circumference=2 π.radius

C=2× 3.14× 3

=6× 3.14=18.84cm

Example2

3cm

What is the circumference of a circle with a radius 5cm?

Circumference=2 π.radius

C=2× 3.14× 5

=10× 3.14

=31.4cm

Example3

A circles circumference is 102 inches. What is the diameter of the circle?

Circumference= π.diameter

102= π.diameter

102/ π= π.diameter/ π

32.5=diameter

EXERCISE

1. What is the circles circumference?

2. What is the circumference of the circle pictured below?

3. A circles circumference is 22 inches. what is the radius of the circle?

4. What is the circumference of the circle pictured below?

7cm

22cm

12cm

Perimeter

Perimeter is the distance around a two dimensional shape, or the measurement of the distance

around something; the length of the boundary.

A perimeter is a path that surrounds a two-dimensional shape. The word comes from the Greek

peri (around) and meter (measure). The term may be used either for the path or its length - it can be

thought of as the length of the outline of a shape. The perimeter of a circle is called its circumference.

Perimeter of a Circle

where I s the radius of the circle and is the diameter.

AREA OF CIRCLE

The area of a circle is all the space inside a circles circumference

In the picture on the above, the area of the circle is

the part of the circle that is red color.

A circle has radius r ,the area of a circle is

Area = π × r2 

r = radiusArea = π × r2 r = radius

The area of a circle is all the space inside a circles circumference.

In the picture on the above, the area of the circle is the part of the

circle that is red color.

Example-4

What is the area of the circle?

3cm

Area= π × r2 

=3.14× 3× 3

=9× 3.14=28.26cm2

Example-5

What is the diameter of a circle if its area is 360cm2?

Area= π × r2 

360=3.14× r2 

r2 =360/3.14=114.591559

r= 114.591559

=10.7047447

Diameter=2r=2×10.7047447=21.40948

EXERCISE

1. What is this circles area?

2. What is the area of a circle with a radius 7 centimeters?

3. What is the radius of a circle .if its area is 120 cm2?

4. A circle has a diameter of 12 inches .What is its area in terms of π?

16cm